Two-loop renormalization of the electric charge in the standard model

We discuss the renormalization of the electric charge at the two-loop level in the standard model of the electroweak interactions. We explicitly calculate the expression of the complete on-shell two-loop counterterm using the background field method and discuss the advantages of this computational approach. We consider the related quantity (e) over cap (2)(mu), defined in the (MS) over bar renormalization scheme and present numerical results for different values of the scale mu. We find that the full two-loop electroweak corrections contribute more than ten parts in units 10(-5) to the Delta(alpha) over cap (m(z)(2)) parameter, obtaining (alpha) over cap (-1)(m(z))=128.12+/-0.05 for Deltaalpha(had)((5))(m(z)(2))=0.027 572+/-0.000 359.